I know that the general solution to a difference equation like y(n) =
Ay(n-1) + By(n-2) is given by y(n) = as^n + bs^n when y is in the form
s^t. But I don't really know why that's the case. Can you explain
where that comes from?

I have a set of data and I'm trying to find a best-fit equation. I've
used linear regression with little success, and I don't know much
about how to do other non-linear regressions. Can you get me started?

I need to find the general equation of the ellipse resulting from the
intersection of an ellipsoid and a plane. When I solve the system of
two equations, I wind up with an xy term and need to know how to
eliminate that from the final equation of the ellipse.

(a) The set Sm = {(2a,b - a,b + a,b) : a,b are real numbers. Under
the operations of matrix addition and multiplication, prove that this
is a subspace of M22. (b) Find a basis for this subspace and give the
dimension of the subspace.